FFF. The F Distribution

As noted several times earlier, variance measures variability. Comparison of variability, therefore, involves comparison of variance. Suppose σ 1 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429124754/be33da89-95f5-45e7-a067-f2fa13b8eb97/content/eq1516.tif"/> and σ 2 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429124754/be33da89-95f5-45e7-a067-f2fa13b8eb97/content/eq1517.tif"/> represent the unknown variances of two independent normal populations. The null hypothesis, H 0 : σ 1 2 = σ 2 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429124754/be33da89-95f5-45e7-a067-f2fa13b8eb97/content/eq1518.tif"/> , asserts that the two populations have the same variance and are therefore characterized by the same variability. When the null hypothesis is true, the test statistic, s 1 2 / s 2 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429124754/be33da89-95f5-45e7-a067-f2fa13b8eb97/content/eq1519.tif"/> , has an F distribution with parameters n ₁ – 1 and n ₂ – 1 called degrees of freedom. Here, s 1 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429124754/be33da89-95f5-45e7-a067-f2fa13b8eb97/content/eq1520.tif"/> is the sample variance of a random sample of n₁ observations from the normal population having variance σ 1 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429124754/be33da89-95f5-45e7-a067-f2fa13b8eb97/content/eq1521.tif"/> , while s 2 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429124754/be33da89-95f5-45e7-a067-f2fa13b8eb97/content/eq1522.tif"/> is the sample variance of a random sample of n ₂ observations from an independent normal population having variance σ 2 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429124754/be33da89-95f5-45e7-a067-f2fa13b8eb97/content/eq1523.tif"/> . When s 1 2 / s 2 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429124754/be33da89-95f5-45e7-a067-f2fa13b8eb97/content/eq1524.tif"/> is used as a test statistic, the critical region depends on the alternative hypothesis, H ₁, and α, the tolerated probability of a Type I error. Listed in Table 27.1 are the critical regions for three different alternative hypotheses. Table 27.1 Critical Regions: <italic>F</italic> Distribution

Alternative Hypothesis (H ₁)

Critical Region

σ 1 2 ≠ σ 2 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429124754/be33da89-95f5-45e7-a067-f2fa13b8eb97/content/eq1525.tif"/>

s 1 2 / s 2 2 < F 1 − α/2  or  s 1 2 / s 2 2 > F α/2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429124754/be33da89-95f5-45e7-a067-f2fa13b8eb97/content/eq1526.tif"/>

σ 1 2 > σ 2 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429124754/be33da89-95f5-45e7-a067-f2fa13b8eb97/content/eq1527.tif"/>

s 1 2 / s 2 2 > F α https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429124754/be33da89-95f5-45e7-a067-f2fa13b8eb97/content/eq1528.tif"/>

σ 1 2 < σ 2 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429124754/be33da89-95f5-45e7-a067-f2fa13b8eb97/content/eq1529.tif"/>

s 1 2 / s 2 2 < F α https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429124754/be33da89-95f5-45e7-a067-f2fa13b8eb97/content/eq1530.tif"/>